Common Core Algebra II.Unit 3.Lesson 5.Inverses of Linear Functions

So in the first problem, it says on the grid below the linear function Y equals two X minus four is graphed along with the line Y equals X let array how can you quickly tell that Y equals two X minus four is a one to one function. Do you remember this? Pause the video for a minute and think about this. Ah, so because it passes, that almost looks like passes. The horizontal line test remember that we had this test that said, and I'm going to get rid of some of these in a little bit. But said, you know, if I draw a horizontal line, remember we're looking at just this line here. If it strikes the function only once, well, then it's a one to one function. And if it doesn't, if it hits it more than once, it's not a one to one function. So it passes the horizontal line test. Let her be says graph the inverse of two X minus four on the same grid. Recall that this can easily be done by switching the X and Y coordinates of the original line. So for instance, this point has coordinates negative one one two three four 5 6 negative 6. 

So the inverse will have coordinates of negative 6, negative one. Right? This one, which is at zero comma negative four, will now be at negative four comma zero. This one, which is at one common negative two, will now be at negative two one. Here, where we have two comma zero. We'll now have zero comma two. Here, three comma two will now be two comma three. This one, which is at four four, stays exactly where it is. And this one, which is at one, two, three, four, 5, 6, will suddenly be at 6 5. So let me let me graph that. Maybe do it well. I guess we'll just do it in blue. All right, and it's going to look something like this. Extend it a little bit this direction too. All right. All right, there's my inverse. What can be said about the graphs of Y equals two X, that's this one. Y equals two X minus four, sorry. And it's inverse Y equals X anything? Anything you can say about them? Well, you can say. They are symmetric. 

That's the word. They are symmetric across Y equals X. Remember what that means is that if I literally flipped this graph over that line, if I folded this graph paper along that line, these two lines would fall on top of each other. Now letter D says find the equation of the inverse in Y equals MX plus B four. What we can do that pretty easily, let's just take a look. It's got a Y intercept of two. It's right here. It slope, well, if I go two to the right and one up, it looks like it's slope, change in Y divided by the change in X will be one to two. So that's going to be Y equals one half X plus two. All right? It's easy enough. Now let me use a little weird. It says, find the equation of the inverse and Y equals X plus B divided by a form. Okay, well, I can actually just do that from this equation. Watch. I'm going to just call that X divided by two. Plus four divided by two, right? I'm just getting a common denominator. So Y will be equal to X plus four divided by two. That's my inverse. 

Now the reason that I want to get to do that, the reason I wanted you to do that was to compare the inverse with the original function compare these two. In this function, we take the input, we multiply by two, and we subtract four. In this one, we add four and we divide by two. Notice how we're doing it in the opposite order. Everything's opposite. Multiply by two subtract four, add four divide by two. So the inverse undoes what the original function does. And in the opposite order, which is really kind of cool. All right. This is going to eventually lead us to how we find the equation of the inverse in a way that's maybe a little bit easier than what we did here. But for now, pause the video and copy down anything you need to. Okay, clear not the text. Exercise number two, which of the following represents the equation of the inverse of Y equals 5 X -20. All right. So let's talk about how you find equations of inverses.

This is a procedure, kind of an algorithm, but it's really cool because all you do is it's really step by step is you switch X and Y so that'll be X equals 5 Y -20, and then you solve for Y but be careful. All right, in fact, let's do it. Let's solve this thing for Y well, I'm going to just rewrite it real quick. X equals 5 Y -20. To solve this for Y, the first thing I'm going to do is add 20 to both sides. I think I'm going to then rewrite it like this X plus 20. And now I'm going to take a look at all those answers. Notice how they all have kind of a one 5th sort of in them. I'm going to put a one 5th on both sides. Now, this is the tricky part to get notice how I kind of threw in those parentheses. It's very important that that one 5th multiplies both the X and the 20. Not just the X and there's our equation. All right. So we switch X and Y and then we carefully solve for Y and that gives us the inverse. The reason that this works is for two reasons. The first reason being that what the inverse does is switch the roles of the input and the output. 

So it kind of makes sense to switch X and Y but then in the process of solving for Y, you literally bring in every inverse operation that the original function or from the original function. So in other words, the original function had a subtracting 20 and multiplying by 5. In order to solve for Y, we had to add 20 and divide by 5. And therefore we get a function that has all the inverse operations, but in the opposite order, which is what you need. All right. Pause the video now and write down anything you need to. That whole issue with inverse functions, whether they're linear or not. Having exactly the opposite operations or inverse operations in the opposite order makes some sense if you think about it. It's what I call the shoes and socks idea. In the morning, I pull my socks on. And then I pull my shoes on. At night when I take them off, I take the shoes off first, and then I take the socks off, right? So I actually have to take the things off in the opposite order in which I put them on. And that makes sense. You do that or I'm going to end up with a sock pulled over a shoe, which would be a little weird. All right. Let's move on to the next exercise. 

Exercise three, which of the following represents the inverse of this linear function. So we are talking about almost exactly the same problem. All right, but with the numbers changed. So why don't you go ahead and challenge this one? All right, this one's a little bit harder because the slope of two thirds is not nearly as nice. But watch as we go through exactly the same process, we're going to say X is equal to two thirds Y plus 8. All right, so first things first, we switch X and Y then we're going to solve for Y now again, we have to be very careful. You know, even more careful than we were in the last problem. Get those parentheses around that. Boom, boom, boom, boom, boom. Distribute the three halves. We'll have Y equals three halves X and then we have to multiply three halves. I don't want to use an X there by 8. So that'll be 12. There it is. Choice two. All right, switch X and Y, carefully solve for Y okay. I'm going to clear this out, so pause the video if you need to. All right, there it goes. Holy cow, we just keep getting it. Number four, what is the Y intercept of the inverse of Y equals three fifths X -9? I need the Y intercept of that inverse. Oh boy. 

Well, lots of different ways to do this. Why don't you pause the video and think about it a little bit? Okay. Well, I suppose the most straightforward way to figure out the Y intercept of the inverse. Would be to come up with the equation of the inverse. In other words, if I say X is equal to three fifths Y -9, add 9 to both sides, multiply both sides by 5 thirds. And we'll get Y equals 5 thirds X now we have to think about what 5 thirds times 9 is. And that's plus 15. So the Y intercept is 15. That's pretty much the most straightforward way of doing it, right? I mean, if I know how to identify the Y intercept by just looking at the equation, then obviously if I can find the equation of the inverse, the Y intercept is not that bad. Okay. I'm going to clear this out so pause the video if you need to. Okay, here we go. It's a multiple choice questions in this lesson, but they're good. They're good to play around with. Number 5, which of the following would be an equation for the inverse of this line, ah, now they gave me a line in point slope form and I want the inverse. Well, the procedure doesn't change. It doesn't matter whether we're dealing with a linear function. 

A linear function of Y equals MX plus B or a linear function in the point slope form, the plain fact is. Step one, I'm going to switch X and Y step two. I'm going to solve for Y now. What's a little bit tricky here? Is it would be very tempting to distribute the four and do a bunch of other manipulation. But take a look at all of these equations. They're all in point slope form. So instead of distributing that four, what I'm going to do is I'm going to multiply both sides of my equation by one fourth. Those two will cancel and believe it or not, I'm done. Right, there it is. Y minus two is equal to X or is equal to one fourth X plus 6. Oh, there it is. Same procedure. We switch X and Y and we solve for Y. All right. I'm going to clear this out. I'm going to try to clear it out. There we go. Okay. Another point slope form. Which of the following points lies on the graph of the inverse of Y -8 equals 5 times X plus two, explain your choice. Well, think about this. There's a lot of different ways to do it. There is one way that's a little bit, let's say, quicker than the others. But play around. All right. Well, I mean, one way of doing it, I'm not going to do it this way. 

I'm going to show you the quicker way. But one way of doing it is actually to come up with the equation of the inverse. Plug in every one of those points and see which one lies on it. All right, but here's what's real easy. We should be able to look at this and say, oh, I can tell that the point negative two 8 lies on that function. Think about it. Y minus Y one equals M times X minus X one. So therefore, X one is negative two and Y one is positive 8. Now, that means the inverse must have these two flip flopped, the inverse must contain the point a comma negative two. And there it is. All right. That's a little tricky because what it means is I've got to be able to look at that point slope form, read off the point negative two comma 8. And then flip them to get the correct point. All right, I'm going to clear this out. Here we go. All right. Let's take a look at one last one. Which of the following linear functions would not have an inverse that is also a function. Explain how you made your choice. There's a lot of good ways of thinking about this, but take a look. One of those linear functions, and they all are linear functions. 

With not have an inverse that's also a function. Let's go through it. Well, the correct answer is choice three. Now, many people might choose choice three because they might look at it and go, well, that's the only one without an X in it. And that's not a bad way to think about it. One way to think about it is that Y equals two beautifully drawn is a horizontal line. Still means it's a linear function. But it's a horizontal line. Therefore, it fails the horizontal line test. Horizontal line test. Therefore, it doesn't have an inverse that's a function. Another way of looking at it is that if I flip flopped, the variables, it's inverse quote unquote, would be X equals two. Definitely not a function. This is a vertical line. Vertical lines are most certainly not functions. All right, so choice three. A little tricky, but it brings in that whole one to one function piece horizontal line test, et cetera. 

All right, I'm going to clear this out, and we'll finish up. So in today's lesson, what I wanted to do is review a lot of the basics about inverse functions about how to graph them, their symmetry across the line Y equals X but also develop the way that we find the formulas for inverse functions. That's going to be very important this year. And if you take pre calculus next year, it'll also be important there. And the method is pretty easy. Switch X and Y, because that's what inverse is due. And then solve for Y all right. Well, thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. And keep solving problems.